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<title>Atlas software user guide -- Parameters</title>
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<h2>Parameters</h2>
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<i>Last updated: October 16, 2005</i>
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Let G be a connected reductive complex group, with a fixed 
<a href="innerclass.html">inner class</a> of
real forms. We are interested in parametrizing isomorphism classes of 
irreducible <a href="gKmod.html">(<b>g</b>,K)-modules</a>, for the various
<a href="realforms.html">real forms</a> in the given inner class.
Let G<sup>&or;</sup> be the <a href="rootdata.html">dual group</a> of G.
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<p>
We refer to Jeff Adams' 2004 <a href="http://atlas.math.umd.edu/papers">
paper</a> for background. Fix a regular integral infinitesimal character
&chi;, and fix a <a href="strongreal.html">&#8220;packet&#8221</a> of strong
real forms of G, corresponding to an element z &#8712; Z(G). Then the problem 
that's stable under duality and affords a nice description is the following:
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<p><i>
Classify isomorphism classes of irreducible <b>g</b>,K)-modules with 
infinitesimal character &chi; for the various strong real forms in the packet 
corresponding to z.
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<p>
The answer is expressed in terms of (G&#215;G<sup>&or;</sup>)-conjugacy classes
of &#8220;sextuples&#8221 in Jeff's paper. However, after reduction, one may 
formulate it as follows. Notice that the categories of
<b>g</b>,K)-modules with infinitesimal character &chi; are equivalent for
values of &chi; that differ by translation by the character lattice of T
(also the cocharacter lattice of T<sup>&or;</sup>); and translation classes
of cocharacters of T<sup>&or;</sup>; are in (1,1)-correspondence with
elements of Z(G<sup>&or;</sup>). So the translation class of &chi;
corresponds to a central element <sup>d</sup>z in G<sup>&or;</sup>, and
therefore also to a packet of strong real forms for G<sup>&or;</sup>.
Denote \cal X(z) the set of <a href="strongreal.html">strong involutions</a>
in N.&delta; with square z, up to T-conjugacy, and define 
\cal X<sup>&or;</sup>(<sup>d</sup>z) analogously (using the 
<a href="rootdata.html">dual inner class</a>). 
Then the set of representations we're after is in natural (1,1)-correspondence
with the set of pairs (&xi;,&eta;) in 
\cal X(z) &#215; \cal X<sup>&or;</sup>(<sup>d</sup>z) such that the 
restrictions &sigma;, &tau; of &xi; and &eta; to T an T<sup>&or;</sup>
respectively satisfy &tau;=-<sup>t</sup>&sigma;.
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<p>
This description has a number of nice features. First of all, since the
conjugacy classes of Cartans are <a href="cartan.html">described</a> in
terms of conjugacy classes of twisted involutions, we have the familiar
partition of representations in terms of Cartans, with discrete series
corresponding to compact tori (if any.) Second, there is an obvious 
(and beautiful!) duality
by reversing the roles of G and G<sup>&or;</sup>: the same picture also
describes the irreducible representations of the packet of strong real forms
of G<sup>&or;</sup> defined by <sup>d</sup>z, for the translation class
of regular integral infinitesimal characters corresponding to z. Third, the 
partition afforded by the various strong real forms of G and G<sup>&or;</sup> 
corresponds to the <a href="blocks.html">blocks</a> in the categories of 
representations with infinitesimal character &chi;, for the strong real forms 
of G in the chosen packet (or dually, to blocks in the corresponding categories
for strong real forms of G<sup>&or;</sup>.)
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<p>
If we are interested in the representations of a fixed real form of G, we
should of course consider only the pairs (&xi;,&eta;) where &xi; belongs
to the G-conjugacy class corresponding to a fixed strong real form mapping to
our chosen real form (after having chosen z so that such a strong real form
exists.)
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